# [EM] Kemeny's Rules = Condorcet's Method

Markus Schulze markus.schulze at alumni.tu-berlin.de
Mon Jan 6 16:02:36 PST 2003

```Dear Steve Barney,

I have found only 4 instances where Condorcet discusses situations
with more than 3 candidates. It seems to me that Condorcet mistakenly
believes (1) that whenever you lock successively the strongest defeats
then you get a unique ranking before a defeat creates a cycle and
(2) that whenever you drop successively the weakest defeats until
you have no cycles anymore then you get a unique ranking.

******

Instance 1 ("Essai sur l'application de l'analyse a la probabilite
des decisions rendues a la pluralite des voix," Imprimerie Royale,
Paris, p. LXVIII, 1785):

> From the considerations we have just made we get the general
> rule that whenever we have to choose we have to take successively
> those propositions that have a plurality - beginning with those
> that have the largest - and to pronounce the result as soon as
> these first propositions create one.

Instance 2 ("Essai sur l'application de l'analyse a la probabilite
des decisions rendues a la pluralite des voix," Imprimerie Royale,
Paris, p. 126, 1785):

> Create an opinion of those n*(n-1)/2 propositions which win
> most of the votes. If this opinion is one of the n*(n-1)*...*2
> possible, then consider as elected that subject to which this
> opinion agrees with its preference. If this opinion is one of the
> (2^(n*(n-1)/2))-n*(n-1)*...*2 impossible opinions, then eliminate
> of this impossible opinion successively those propositions that
> have a smaller plurality & accept the resulting opinion of the
> remaining propositions.

Instance 3 ("Sur la Forme des Elections," 1789):

> To compare just 20 candidates two by two, we must examine
> the votes on 190 propositions, and for 40 candidates, on 780
> propositons. Besides, this will often give us an unsatisfactory
> result; it may be that no candidate is considered by the
> plurality to be better than all the others, and then we would
> have to prefer the candidate who is just considered better
> than a larger number; and when several were considered better
> than the same number of candidates, we would have to choose
> the candidate who was either considered better by the greatest
> plurality, or worst by the smallest plurality. However, this
> preference is sometimes difficult to determine: the general
> rule would be complicated and awkward to put into practice.
> This form of election is therefore only really suitable when
> we do not need to make an immediate choice, or when new electors
> can be quickly summoned if the elections is undecided; moreover,
> even this last solution does not assure success, but makes it
> more probable.

Instance 4 ("Sur les Elections," Journal d'Instruction Sociale,
vol. 1, pp. 25-32, 1793):

> A table of majority judgements between the candidates
> taken two by two would then be formed and the result - the
> order of merit in which they are placed by the majority -
> extracted from it. If these judgements could not all exist
> together, then those with the smallest majority would be
> rejected.

Markus Schulze

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